Robust Adaptive Synchronization of a Class of Uncertain Chaotic Systems with Unknown Time-Delay

Assef Zare,Roohallah Alizadehsani,Mahboobeh Jafari,Majid Hallaji,Amir Mosavi,Afshin Shoeibi,Seyede Zeynab Mirrezapour,Navid Ghassemi

doi:10.3390/app10248875

Abstract

In this paper, a robust adaptive control strategy is proposed to synchronize a class of uncertain chaotic systems with unknown time delays. Using Lyapunov theory and Lipschitz conditions in chaotic systems, the necessary adaptation rules for estimating uncertain parameters and unknown time delays are determined. Based on the proposed adaptation rules, an adaptive controller is recommended for the robust synchronization of the aforementioned uncertain systems that prove the robust stability of the proposed control mechanism utilizing the Lyapunov theorem. Finally, to evaluate the proposed robust and adaptive control mechanism, the synchronization of two Jerk chaotic systems with finite non-linear uncertainty and external disturbances as well as unknown fixed and variable time delays are simulated. The simulation results confirm the ability of the proposed control mechanism in robust synchronization of the uncertain chaotic systems as well as to estimate uncertain and unknown parameters.

Highlights

The dynamic behaviors of a chaotic system looks like a random behavior
Robust synchronization of the master and slave chaotic systems with uncertainty and unknown time delays described in forms (1) and (2) is fulfilled if the controller u0 (t) characterized in form (8) satisfies the condition of Theorem 1 and the adaptive controller u(t) is designed as follows: u(t) = f (x(t − τ1 )) − g( y(t − τ2 )) − sign(en )(γ1 + γ2 + α1 h1 (x) + α2 h2 ( y)) − k0 en, (11)
Applying the adaptive control strategy proposed in the previous section, the robust synchronization of two Jerk chaotic systems with uncertainty and bounded external disturbances as well as unknown time delays is simulated and evaluated

Summary

Introduction

The dynamic behaviors of a chaotic system looks like a random behavior. But in reality, the behavior of chaotic systems follows a natural order. In [19], the design of a synchronization controller based on robust adaptive neural networks for a class of time-delay uncertain chaotic systems is yielded. This paper deals with an adaptive control strategy for robust synchronization of uncertain chaotic systems with unknown fixed and variable time delay. In order to evaluate the performance of the proposed control strategy, the synchronization of two Jerk chaotic systems with unknown fixed and variable time-delay coupled with uncertainty and disturbance are investigated and simulated. Adaptive controller design to robust synchronization of fractional order chaotic systems with unknown time-delay, uncertainty and disturbance is givenin Section 3. The synchronization of two Jerk chaotic systems with unknown fixed and variable time delay and uncertainty and disturbance is analyzed and simulated, and in Section 5, conclusions based on the stated theories are presented The synchronization of two Jerk chaotic systems with unknown fixed and variable time delay and uncertainty and disturbance is analyzed and simulated in Section 4, and in Section 5, conclusions based on the stated theories are presented

Preliminaries and Problem Formulation

Simulation Example and Results

Conclusions